Tail Dependence from a Distributional Point of View

The dependence structure in the tails of bivariate random variables is studied by means of appropriate copulae. Weak convergence results show that these copulae are natural dependence structures for joint tail events. The results obtained apply to particular types of copulae such as archimedean copulae and the Gaussian copula. Further, connections to multivariate extreme value theory are investigated and a two-dimensional Pickands–Balkema–de Haan Theorem type is derived. Finally, a counterexample showing that the tail dependence coefficients do not completely determine the dependence structure of bivariate rare events is provided.     

Tail dependence for elliptically contoured distributions

The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.     

A copula entropy approach to correlation measurement at the country level

The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.     

Extreme value properties of multivariate t copulas

The extremal dependence behavior of t copulas is examined and their extreme value limiting copulas, called the t-EV copulas, are derived explicitly using tail dependence functions. As two special cases, the Hüsler–Reiss and the Marshall–Olkin distributions emerge as limits of the t-EV copula as the degrees of freedom go to infinity and zero respectively. The t copula and its extremal variants attain a wide range in the set of bivariate tail dependence parameters. Supported by NSERC Discovery Grant.     

Dependence Graphs: Dependence Within and Between Groups

This paper applies the two-party dependence theory (Castelfranchi, Cesta and Miceli, 1992, in Y. Demazeau and E. Werner (Eds.) Decentralized AI-3, Elsevier, North Holland) to modelling multiagent and group dependence. These have theoretical potentialities for the study of emerging groups and collective structures, and more generally for understanding social and organisational complexity, and practical utility for both social-organisational and agent systems purposes. In the paper, the dependence theory is extended to describe multiagent links, with a special reference to group and collective phenomena, and is proposed as a framework for the study of emerging social structures, such as groups and collectives. In order to do so, we propose to extend the notion of dependence networks (applied to a single agent) to dependence graphs (applied to an agency). In its present version, the dependence theory is argued to provide (a) a theoretical instrument for the study of social complexity, and (b) a computational system for managing the negotiation process in competitive contexts and for monitoring complexity in organisational and other cooperative contexts.     

Dynamic Bivariate Mortality Modelling

The dependence structure of the life statuses plays an important role in the valuation of life insurance products involving multiple lives. Although the mortality of individuals is well studied in the literature, their dependence remains a challenging field. In this paper, the main objective is to introduce a new approach for analyzing the mortality dependence between two individuals in a couple. It is intended to describe in a dynamic framework the joint mortality of married couples in terms of marginal mortality rates. The proposed framework is general and aims to capture, by adjusting some parametric form, the desired effect such as the “broken-heart syndrome”. To this end, we use a well-suited multiplicative decomposition, which will serve as a building block for the framework to relate the dependence structure and the marginals, and we make the link with existing practice of affine mortality models. Finally, given that the framework is general, we propose some illustrative examples and show how the underlying model captures the main stylized facts of bivariate mortality dynamics.

     

The Computer Simulation of Partnership Formation

This paper is intended to present and apply a formal dependence model to a simulation study of partnership formation. Although the emergence and evolution of coalitions is an issue of major concern in the study of organisations, a number of questions are still laid open. How do coalitions emerge? Which processes and mechanisms are responsible for their evolution? The agents&;#x2018; informal communications, commitments and negotiations are considered as &;#x201C;given&;#x201D;, and none or poor effort is done to ground them upon the agents&;#x2018; self-interests. The philosophy underlying this paper is that the objective relationships of dependence among heterogeneous agents provide a fundamental ground for the emergence of spontaneous coalitions. As long as agents are endowed with different goals and heterogeneous competencies, a structure of social relationships, namely dependence relations, is likely to occur among them. A formal model of dynamic dependence relationships based on the agents&;#x2018; individual properties will be used to derive a further agents&;#x2018; property, namely their negotiation powers. Through computer simulation, this property will be used to predict how likely each agent in a population will form rewarding partnerships.     

Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.     

Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein–Uhlenbeck processes

While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a Lévy-driven spatio-temporal Ornstein–Uhlenbeck process by randomly varying its rate parameter to model both short-range and long-range dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We develop a simulation algorithm for the compound Poisson case which can be used to approximate other Lévy bases. The generalized method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties.     

Two-dimensional inverse heat conduction problem of source strength estimation in cylindrical rods

The radial and circumferential (azimuthal) transient dependence of the strength of a volumetric heat source in a cylindrical rod is estimated with Alifanov's iterative regularization method. This inverse problem is solved as an optimization problem in which a squared residue functional is minimized with the conjugate gradient method. A sensitivity problem is used in the determination of the step size in the direction of descent, while an adjoint problem is solved to determine the gradient. In order to examine the accuracy of estimations, two test cases are considered, one with a radial and timewise dependence and the second with radial, azimuthal as well as timewise dependence. The effects of number of sensors and measurement errors are investigated.     

Tail dependence functions and vine copulas

Tail dependence and conditional tail dependence functions describe, respectively, the tail probabilities and conditional tail probabilities of a copula at various relative scales. The properties as well as the interplay of these two functions are established based upon their homogeneous structures. The extremal dependence of a copula, as described by its extreme value copulas, is shown to be completely determined by its tail dependence functions. For a vine copula built from a set of bivariate copulas, its tail dependence function can be expressed recursively by the tail dependence and conditional tail dependence functions of lower-dimensional margins. The effect of tail dependence of bivariate linking copulas on that of a vine copula is also investigated.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

Technology and xx

The revolution of technology in education has heralded a new wave of learning styles and in some cases total dependence on technology. Yet, statistics reveal no significant improvement in student performance in Mathematics and a downward trend in basic algebra skills. The major impediment in the learning process is lack of understanding of the concepts and a virtual total dependence on technology. We expose the dangers of total dependence on technology by using the much-ignored function f(x) = x^x as a case study and conclude that technology can be a valuable tool provided the student has complete understanding of related concepts.     

Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

The structure of various Gerber-Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg’s fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.     

On the differences and similarities of the first order delay and ordinary differential equations

The present paper is the first step in the systematic study of differences and similarities of the first order delay and ordinary differential equations. The continuous dependence of solutions to DDE on the time delay tending to zero is discussed and theorems guaranteeing continuous dependence are proved. The properties of nonnegativity and the blow-up phenomena for the solution to delay differential equation are studied. Conditions guaranteeing boundness of the solution to DDE are stated. Delay differential equations are considered in Rn as well as in Lp.     

Central Regions and Dependency

The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.     

Nonparametric Bayesian Assessment of the Order of Dependence for Binary Sequences

This article discusses inference on the order of dependence in binary sequences. The proposed approach is based on the notion of partial exchangeability of order k. A partially exchangeable binary sequence of order k can be represented as a mixture of Markov chains. The mixture is with respect to the unknown transition probability matrix θ. We use this defining property to construct a semiparametric model for binary sequences by assuming a nonparametric prior on the transition matrix θ. This enables us to consider inference on the order of dependence without constraint to a particular parametric model. Implementing posterior simulation in the proposed model is complicated by the fact that the dimension of θ changes with the order of dependence k. We discuss appropriate posterior simulation schemes based on a pseudo prior approach. We extend the model to include covariates by considering an alternative parameterization as an autologistic regression which allows for a straightforward introduction of covariates. The regression on covariates raises the additional inference problem of variable selection. We discuss appropriate posterior simulation schemes, focusing on inference about the order of dependence. We discuss and develop the model with covariates only to the extent needed for such inference.     

Approximation of bivariate copulas by patched bivariate Fréchet copulas

Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums.     

A measure of multivariate mutual complete dependence

The authors propose a multivariate version of Siburg and Stoimenov’s measure of mutual complete dependence. This multivariate version is, however, not the distance between a copula and the product copula $C_I$ under the modified Sobolev norm since the set of mutual complete dependence copulas does not lie on the sphere centered at $C_I$. To overcome this difficulty, the authors choose another center and define measures of complete dependence based on the modified Sobolev norm and this center. The measure of multivariate mutual complete dependence is then defined as the summation of the (normalized) measures of complete dependence.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.